"DNS server not responding"

Check the wireless router settings to make sure it is not trying to assign DHCP IP addresses and/or DNS IP addresses.
All that should be happening is that your wireless router gives you your local network connection and then your IP address, DNS and Gateway addresses should all come from the broadband router.
If the broadband device is just a modem, then I have to assume that you have a separate router to which the modem is connected. In which case the router has to be configured in the correct way to route all Internet traffic to the modem.

The DNS settings (which you may have to specify as part of the DHCP configuration) ought to match the DNS servers which your broadband provider stipulates should be used. A lot of ISP's now block DNS requests from IP addresses which are not part of their IP address network.

FYI:
DNS servers are vital to Internet functionality. It is the DNS servers which translate the URL, say, www.google.com into the IP address which is actually required to get to the relevant server.